Info

[7.19,20]

Melting temperature

1180°C

[7.21]

Sublimation temperature

434° C

[7.22]

Heat of sublimation

40.1 kcal/mol

[7.22]

Latent heat

1.65 eV/Cjo

[7.23]

"The bulk modulus is 6.8 GPa at room temperature in the fee phase and 8.8 GPa in the sc phase below r01 [7.5]. Earlier reported values for the bulk modulus were in the range 14-18 GPa [7.4,24-26], The conversion of units for pressure measurements is 1 GPa = 109 N/m2 = 0.99 kbar = 7.52 x 106 torr = 1010 dyn/cm2. Also, 1 atm = 760

6For solvated C60 crystals measured by vibrating reed method [7.7].

Measurements on film samples suggest a higher Young's modulus

Structural phase transition discussed in §7.1.2.

C60 should increase significantly, as discussed in §7.3. A proposed phase diagram for C60 [7.28] is shown in Fig. 7.1 [7.9,28-31], As the temperature and pressure are both increased, two distinct dense phases of Cm have been identified (see §7.3). In this section, we first describe the crystal structure of C60 at room temperature and then review the temperature dependence of the crystal structure.

109 107 105 103 I 10'

3 10"

100 1000 10000 Temperature (K)

Fig. 7.1. Phase diagram for C60. Solid lines are from experiment and dashed lines are theoretical or reflect extrapolations from experiment. The pressure dependence of the transition from the simple cubic (sc) to the face-centered cubic (fee) phases is from Ref. [7.9] and the delineation between the fee and vapor phase is based on the results of Refs. [7.28-30], The data at high temperature and pressure reflect calculated boundaries between the liquid, solid, and vapor phases from Ref. [7.31], The existence of a liquid phase remains to be proven. Molecular collapse and fragmentation are indicated at high pressure or high temperature [7.28].

7.1.1. Ambient Structure

At room temperature, the Cgg molecules in the pristine solid have been shown by nuclear magnetic resonance (NMR) [7.32-36] to be rotating rapidly with three degrees of rotational freedom. The molecular centers themselves are arranged on a face-centered cubic (fee) lattice (see Fig. 7.1) with one C60 molecule per primitive fee unit cell, or four molecules per simple cubic unit cell (see Fig. 7.2). Since the molecules are spinning rapidly about their lattice positions, there is no orientational order, and all molecules are equivalent in measurements which require a time longer than the average rotational period (t » 10"11 s). Thus the pertinent space group for the structure where the molecules are rotating rapidly is 0\ or Fm3m, using the Schoenflies and International Crystallographic nomenclature, respectively. The symmetry of the space group for C60 at room temperature has been established directly by x-ray and neutron diffraction

109 107 105 103 I 10'

3 10"

100 1000 10000 Temperature (K)

Fig. 7.1. Phase diagram for C60. Solid lines are from experiment and dashed lines are theoretical or reflect extrapolations from experiment. The pressure dependence of the transition from the simple cubic (sc) to the face-centered cubic (fee) phases is from Ref. [7.9] and the delineation between the fee and vapor phase is based on the results of Refs. [7.28-30], The data at high temperature and pressure reflect calculated boundaries between the liquid, solid, and vapor phases from Ref. [7.31], The existence of a liquid phase remains to be proven. Molecular collapse and fragmentation are indicated at high pressure or high temperature [7.28].

Fig. 7.2. Basal plane projection of the low-temperature crystal structure of solid Caj (space group Pa3) [7.38],

Fig. 7.2. Basal plane projection of the low-temperature crystal structure of solid Caj (space group Pa3) [7.38],

[7.11,39,40]. The crystal structure pertinent to the room temperature ambient phase persists down to a temperature T01 where a transition to a simple cubic (sc) phase occurs (see Fig. 7.1 and §7.1.3).

The C-C bond lengths in the solid phase are essentially the same as in the gas-phase (see §3.1). An accurate measurement of the bond length difference between the single (a5) and double bonds (a6) has been carried out by measuring the neutron scattering intensity over a large momentum transfer range (6.5 < Q < 20 A ), over which only the intramolecular structure factor contributes to the scattering [7.41]. In modeling the scattering results, the C60 molecule is considered as a truncated icosahedron with (60)(59)/2 = 1770 pairs of carbon atoms and 23 distinct interatomic distances. The results show a difference in bond lengths a5 — a6 = 0.062 A at 295 K [7.41],

More detailed analysis of single crystal x-ray diffraction data shows that there are some orientational correlations between adjacent C60 molecules even at room temperature [7.42,43], consistent with a model for rapid ratcheting motion, rather than constant angular momentum rotations. The rotational reorientation time at room temperature is 9-12 ps based on NMR [7.32,33,35,36,44], and /j, meson spin resonance (/u-SR) [7.45] meai-surements (see §16.1.4 and §16.3). It has also been suggested that low-frequency elastic relaxation (ratcheting motion) may originate from the appearance and growth of small ordered clusters of a low-symmetry phase (i.e., order parameter fluctuations) [7.46].

Relative to the other allotropic forms of carbon, solid C60 is relatively compressible, with an isothermal volume compressibility (see Table 7.1) of 6.9 x 10"12 cm2/dyn [7.4], approximately three times larger than that of graphite, because the van der Waals charge cloud around the C60 molecules can be compressed easily in three dimensions, rather than in one dimension as in the case of graphite.

It should be noted that the cubic structure described in this chapter is pertinent to single crystals grown from the vapor sublimation of C60 molecules. For single crystals prepared from a solution involving organic solvents, other crystal structures may be stabilized by solvent molecules incorporated into the structure. For example, C60 crystals grown from CS2 solution crystallize in a black crystal with an orthorhombic structure (space group Pbnm) and lattice constants, a = 25.012 A, b = 25.583 A, and c = 10.003 A, and with a unit cell volume of 6401 A3 [7.47], On heating these orthorhombic single crystals above 100°C in vacuum, a transformation is made to the fee structure obtained from vapor sublimation growth. Two different high-pressure phases have also been identified and are discussed on §7.3, but neither of these phases is included in the phase diagram of Fig. 7.1.

7.1.2. Group Theory for Crystalline Phases

In this section we consider the symmetry aspects of solid-state effects associated with the high-temperature and low-temperature phases of crystalline C60. In discussing the symmetry properties of crystalline C60, we must bear in mind that some properties are most sensitive to the symmetry of the constituent molecules (such as the vibrational spectra), while other properties are sensitive to crystalline properties (such as transport phenomena). The major problem to be addressed is the determination of the symmetry elements common to both the icosahedral symmetry of C60 and the crystal field symmetry of the crystal lattice.

Regarding the high-temperature fee phase in Fig. 7.1, there are three possible cases to consider. In one case, the crystal symmetry is emphasized relative to the molecular symmetry, and in the other two cases, the molecular symmetry is emphasized relative to the crystal symmetry. The first case would be appropriate for phenomena where the molecular reorientation is so rapid that the physical property under consideration senses only spherical balls located at fee lattice sites (Fm3m space group or Oh point group symmetry). This first case is seldom considered in the literature, because of the dominance of the molecular vibrational and electronic states in most of the observed physical properties of fullerenes in the crystalline phase. Thus, most treatments of crystalline C60 consider an arrangement of icosa-

hedral C60 molecules relative to the crystal field coordinates, in one of two cases, as outlined below.

The simplest case to consider is the one where all the molecules are similarly oriented with respect to the same crystalline axes, leading to a space group Fm3 (or with the maximum point group symmetry Th and four identical C60 molecules per fee cubic unit cell. The other case is one where the molecules are merohedrally disordered with respect to common crystalline axes, as discussed in §7.1.4 and §8.5.1. If the C60 molecules are randomly oriented, there is no special site symmetry. It should be mentioned that 7m3 is the space group that maximizes the common symmetry operations between the icosahedral molecule (Ih) and the body-centered cubic (bcc) crystal lattice, as occurs for KgQo, Rb6C60, and Cs6C50. In the case of Cs6C60 all the C60 anions have the same orientation of their twofold axes (see §8.5.2).

We first discuss the symmetry lowering due to the placement of icosahedral molecules in the Fm3 fee lattice, preserving all the symmetry elements common between the icosahedrons and the fee lattice; this is represented by the space group Fm3 (see Table 7.2) and the point group Th (m3). This is an fee lattice with four ordered molecules per cubic unit cell. Various other space group symmetries with the twofold axis of the C60 aligned with respect to the (100) crystalline axes are discussed by Harris and Sachi-

Table 7.2

Symmetry sites for the face centered cubic space group (or Fm3) [7.49],

Table 7.2

Symmetry sites for the face centered cubic space group (or Fm3) [7.49],

No.

Site

Sym.

Site coordinates"

(0,0,0; 0,1

i; |,0,i; |,|,0) + listed sites

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